This particular combination of arms threw me a little, until the book that I told you about before, The Heraldry of Ely Cathedral, helped to explain to to me.
But first, let me show you what confused me.
You may wish to click on the image above to see a larger picture that shows the detail of the two shields better.
The initially confusing part to me was the same coat of arms on the sinister side (to the viewer's right) on each of these two shields. I mean, the shield on the left was pretty clearly an ecclesiastical impalement of the arms of the See of Ely with one of it's bishops. But the arms of the bishop would not also appear on the sinister side of an impaled coat of arms where the dexter arms were not those of a See.
But just a little bit of research brought enlightenment!
The shield on the left is the See of Ely (Gules three crowns or) impaling James Yorke, Bishop of Ely 1781-1808 (Argent on a saltire azure a bezant). And the shield on the right the arms of Thomas Waddington, Canon of Ely (Argent a chevron between three martlets gules) impaling Margaret Yorke, the eldest daughter of Bishop James Yorke (Argent on a saltire azure a bezant).
So the Yorke arms in the two shields refer to two different individuals, not one, and that was wherein my confusion lay. (It is widely accepted usage in England, and a number of other places, for a daughter to use her father's arms, even in a marital achievement like the one on the right, above.)
And thus was my initial question of "What exactly is going on here?" was answered quickly and concisely, and it will no longer keep me awake at night wondering about it.
Problem solved! Time for a good night's rest.